#394087

order	512
nilpotency class	3
isoclinism	isoclinic class of #394086
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14715
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3894 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,13)(4,9)(5,10)(6,23)(7,12)(8,16)(11,26)(14,38)(15,39)(17,36)(18,46)(19,37)(20,33)(21,49)(22,35)(24,29)(25,32)(27,45)(28,43)(30,41)(31,48)(34,52)(40,61)(42,54)(44,58)(47,57)(50,55)(51,59)(53,62)(56,64)(60,63);(1,3,7,17)(2,8,12,29)(4,14,20,42)(5,15,22,44)(6,24,26,16)(9,30,33,27)(10,28,35,55)(11,36,23,13)(18,40,48,63)(19,45,25,41)(21,50,52,43)(31,56,46,53)(32,54,37,38)(34,58,49,39)(47,64,51,62)(57,61,59,60);(1,4)(2,9)(3,14)(5,18)(6,25)(7,20)(8,27)(10,31)(11,37)(12,33)(13,38)(15,40)(16,45)(17,42)(19,26)(21,51)(22,48)(23,32)(24,41)(28,53)(29,30)(34,59)(35,46)(36,54)(39,60)(43,64)(44,63)(47,52)(49,57)(50,62)(55,56)(58,61);(1,5)(2,10)(3,15)(4,18)(6,21)(7,22)(8,28)(9,31)(11,34)(12,35)(13,39)(14,40)(16,43)(17,44)(19,47)(20,48)(23,49)(24,50)(25,51)(26,52)(27,53)(29,55)(30,56)(32,57)(33,46)(36,58)(37,59)(38,60)(41,62)(42,63)(45,64)(54,61)

Polycyclic presentation:

2^9|b^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,e^c=ei,h^c=hi,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260677, #72280689, #72282605, …
maximal normal subgroup of	#72260677, #72280689, #72282605, …
maximal quotient of	#72260755

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301283);